Chapter 3 Geometry of convex functions - Meboo Publishing ...

3.6. EPIGRAPH, SUBLEVEL SET 237
confined to the entire positive semidefinite cone (including its boundary). It
is now our goal to incorporate f into an optimization problem such that
an optimal solution returned always comprises a projection matrix W . The
set of orthogonal projection matrices is a nonconvex subset of the positive
semidefinite cone. So f cannot be convex on the projection matrices, and
its equivalent (for idempotent W )
f(W , x) = x T( I − (1 + ǫ) −1 W ) x (550)
cannot be convex simultaneously in both variables on either the positive
semidefinite or symmetric projection matrices.
Suppose we allow domf to constitute the entire positive semidefinite
cone but constrain W to a Fantope (90); e.g., for convex set C and 0 < k < n
as in
minimize ǫx T (W + ǫI) −1 x
x∈R n , W ∈S n
subject to 0 ≼ W ≼ I
(551)
trW = k
x ∈ C
Although this is a convex problem, there is no guarantee that optimal W is
a projection matrix because only extreme points of a Fantope are orthogonal
projection matrices UU T .
Let’s try partitioning the problem into two convex parts (one for x and
one for W), substitute equivalence (548), and then iterate solution of convex
problem
minimize x T (I − (1 + ǫ) −1 W)x
x∈R n
(552)
subject to x ∈ C
with convex problem
(a)
minimize x ⋆T (I − (1 + ǫ) −1 W)x ⋆
W ∈S n
subject to 0 ≼ W ≼ I
trW = k
≡
maximize x ⋆T Wx ⋆
W ∈S n
subject to 0 ≼ W ≼ I
trW = k
(553)
until convergence, where x ⋆ represents an optimal solution of (552) from
any particular iteration. The idea is to optimally solve for the partitioned
variables which are later combined to solve the original problem (551).
What makes this approach sound is that the constraints are separable, the